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arXiv · 2608.06094

Time-Dependent Hamiltonian Simulation with Optimal Query Complexity

Boyang Chen·Minbo Gao·Xinzhao Wang·Shuo Zhou
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Abstract

We give a query-optimal algorithm for simulating a general $n$-qubit time-dependent Hamiltonian $H(t)$ on $[0,T]$, assuming that $H$ is Lipschitz continuous and $\|H(t)\|\leq\alpha$. In the standard $\mathrm{HAM\mbox{-}T}$ access model, the algorithm approximates the time-ordered propagator $U_H(T)$ to error $\varepsilon$ using $$ O\left( \alpha T+\frac{\log(1/\varepsilon)} {\log(e+\log(1/\varepsilon)/(\alpha T))} \right) $$ $\mathrm{HAM\mbox{-}T}$ queries. This matches the known query lower bound for time-independent Hamiltonians, showing that time dependence incurs no asymptotic query overhead. Our method first constructs a one-query transducer that, given an auxiliary state, implements an approximation to $U_H(T)$ and returns the state unchanged. A weighted combination of circuits that apply the transducer different numbers of times makes the error caused by omitting this state decay factorially, yielding the stated optimal precision dependence. For time-independent Hamiltonians, the same method also gives a query-optimal alternative to qubitization.

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Boyang Chen, Minbo Gao, Xinzhao Wang, Shuo Zhou. 2026-08-06. Time-Dependent Hamiltonian Simulation with Optimal Query Complexity. https://arxiv.org/abs/2608.06094

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